Given the function f, of, x, equals, x, to the power 4 , plus, 8, x, cubed, plus, 16, x, squared, plus, 8, comma*f*(*x*)=*x*4+8*x*3+16*x*2+8, determine the absolute maximum value of f*f* on the closed interval open square bracket, minus, 5, comma, minus, 1, close square bracket, .[−5,−1].

To determine the absolute maximum value of the function \( f(x) = x^4 + 8x^3 + 16x^2 + 8 \) on the closed interval \([-5, -1]\), we need to follow these steps:

1. **Find the derivative of \( f(x) \):**
   \[
   f'(x) = 4x^3 + 24x^2 + 32x
   \]

2. **Set \( f'(x) = 0 \) to find critical points within the interval:**

   We solve:
   \[
   4x^3 + 24x^2 + 32x = 0
   \]

   Factor out \(4x\):
   \[
   4x(x^2 + 6x + 8) = 0
   \]
   Then, further factor \( x^2 + 6x + 8 \):
   \[
   4x(x + 4)(x + 2) = 0
   \]

   This gives the solutions:
   \[
   x = 0, \quad x = -4, \quad x = -2
   \]

   Since we are only interested in values within the interval \([-5, -1]\), we discard \( x = 0 \) and keep \( x = -4 \) and \( x = -2 \).

3. **Evaluate \( f(x) \) at the endpoints and critical points in the interval:**

   - At \( x = -5 \):
     \[
     f(-5) = (-5)^4 + 8(-5)^3 + 16(-5)^2 + 8 = 625 - 1000 + 400 + 8 = 33
     \]

   - At \( x = -1 \):
     \[
     f(-1) = (-1)^4 + 8(-1)^3 + 16(-1)^2 + 8 = 1 - 8 + 16 + 8 = 17
     \]

   - At \( x = -4 \):
     \[
     f(-4) = (-4)^4 + 8(-4)^3 + 16(-4)^2 + 8 = 256 - 512 + 256 + 8 = 8
     \]

   - At \( x = -2 \):
     \[
     f(-2) = (-2)^4 + 8(-2)^3 + 16(-2)^2 + 8 = 16 - 64 + 64 + 8 = 24
     \]

4. **Determine the absolute maximum:**

   By comparing these values, we see that the maximum value is:
   \[
   \max\{33, 17, 8, 24\} = 33
   \]

### Conclusion:
The absolute maximum value of \( f(x) \) on the interval \([-5, -1]\) is \( \boxed{33} \), which occurs at \( x = -5 \).

Would you like details on any step, or do you have further questions?

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Here are some related questions you might find useful:
1. How do you determine if a critical point is a maximum or minimum?
2. How would you analyze this function on a different interval?
3. What is the difference between absolute and relative maxima?
4. How can you use the second derivative test to confirm a maximum?
5. What are other methods for solving polynomial equations like this one?

**Tip:** Always evaluate your function at both critical points and endpoints when looking for absolute extrema on a closed interval.

Given the function f(x) = x^4 + 8x^3 + 16x^2 + 8, determine the absolute maximum value of f on the closed interval [-5, -1].

Determine the absolute maximum of f(x) = x^4 + 8x^3 + 16x^2 + 8 on the interval [-5, -1] using calculus concepts. This solution includes finding the derivative, setting critical points, and evaluating endpoints. Suitable for high school calculus students.

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